Theorem nb3grprlem1 29961

Description: Lemma 1 for nb3grpr 29963. (Contributed by Alexander van der Vekens, 15-Oct-2017.) (Revised by AV, 28-Oct-2020.)

Hypotheses

Ref	Expression
nb3grpr.v	⊢ 𝑉 = (Vtx‘𝐺)
nb3grpr.e	⊢ 𝐸 = (Edg‘𝐺)
nb3grpr.g	⊢ (𝜑 → 𝐺 ∈ USGraph)
nb3grpr.t	⊢ (𝜑 → 𝑉 = {𝐴, 𝐵, 𝐶})
nb3grpr.s	⊢ (𝜑 → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍))

Assertion

Ref	Expression
nb3grprlem1	⊢ (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))

Proof of Theorem nb3grprlem1

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nb3grpr.s	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍))
2		prid1g 4721	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑌 → 𝐵 ∈ {𝐵, 𝐶})
3	2	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐵 ∈ {𝐵, 𝐶})
4	1, 3	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ {𝐵, 𝐶})
5	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ {𝐵, 𝐶})
6		eleq2 2850	. . . . . . 7 ⊢ ({𝐵, 𝐶} = (𝐺 NeighbVtx 𝐴) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
7	6	eqcoms 2769	. . . . . 6 ⊢ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
8	7	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
9	5, 8	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ (𝐺 NeighbVtx 𝐴))
10		nb3grpr.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ USGraph)
11		nb3grpr.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
12	11	nbusgreledg 29934	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐵, 𝐴} ∈ 𝐸))
13		prcom 4693	. . . . . . . . 9 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
14	13	a1i 11	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → {𝐵, 𝐴} = {𝐴, 𝐵})
15	14	eleq1d 2846	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → ({𝐵, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
16	12, 15	bitrd 282	. . . . . 6 ⊢ (𝐺 ∈ USGraph → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
17	10, 16	syl 18	. . . . 5 ⊢ (𝜑 → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
18	17	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
19	9, 18	mpbid 235	. . 3 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐵} ∈ 𝐸)
20		prid2g 4722	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑍 → 𝐶 ∈ {𝐵, 𝐶})
21	20	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 ∈ {𝐵, 𝐶})
22	1, 21	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ {𝐵, 𝐶})
23	22	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ {𝐵, 𝐶})
24		eleq2 2850	. . . . . . 7 ⊢ ({𝐵, 𝐶} = (𝐺 NeighbVtx 𝐴) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
25	24	eqcoms 2769	. . . . . 6 ⊢ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
26	25	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
27	23, 26	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ (𝐺 NeighbVtx 𝐴))
28	11	nbusgreledg 29934	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐶, 𝐴} ∈ 𝐸))
29		prcom 4693	. . . . . . . . 9 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
30	29	a1i 11	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → {𝐶, 𝐴} = {𝐴, 𝐶})
31	30	eleq1d 2846	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → ({𝐶, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸))
32	28, 31	bitrd 282	. . . . . 6 ⊢ (𝐺 ∈ USGraph → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
33	10, 32	syl 18	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
34	33	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
35	27, 34	mpbid 235	. . 3 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐶} ∈ 𝐸)
36	19, 35	jca 521	. 2 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))
37		nb3grpr.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
38	37, 11	nbusgr 29930	. . . . 5 ⊢ (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐴) = {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
39	10, 38	syl 18	. . . 4 ⊢ (𝜑 → (𝐺 NeighbVtx 𝐴) = {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
40	39	adantr 486	. . 3 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝐺 NeighbVtx 𝐴) = {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
41		nb3grpr.t	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 = {𝐴, 𝐵, 𝐶})
42		eleq2 2850	. . . . . . . . . 10 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
43	41, 42	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
44	43	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
45		vex 3455	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
46	45	eltp 4650	. . . . . . . . . 10 ⊢ (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑣 = 𝐴 ∨ 𝑣 = 𝐵 ∨ 𝑣 = 𝐶))
47	11	usgredgne 29787	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ USGraph ∧ {𝐴, 𝑣} ∈ 𝐸) → 𝐴 ≠ 𝑣)
48		df-ne 2957	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ≠ 𝑣 ↔ ¬ 𝐴 = 𝑣)
49		pm2.24 125	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 = 𝑣 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
50	49	eqcoms 2769	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐴 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
51	50	com12 33	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝐴 = 𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
52	48, 51	sylbi 220	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ≠ 𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
53	47, 52	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USGraph ∧ {𝐴, 𝑣} ∈ 𝐸) → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
54	53	ex 418	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ USGraph → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
55	10, 54	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
56	55	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
57	56	com3r 88	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐴 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
58		orc 881	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐵 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))
59	58	2a1d 27	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐵 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
60		olc 882	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐶 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))
61	60	2a1d 27	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐶 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
62	57, 59, 61	3jaoi 1454	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐴 ∨ 𝑣 = 𝐵 ∨ 𝑣 = 𝐶) → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
63	46, 62	sylbi 220	. . . . . . . . 9 ⊢ (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
64	63	com12 33	. . . . . . . 8 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
65	44, 64	sylbid 243	. . . . . . 7 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ 𝑉 → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
66	65	impd 416	. . . . . 6 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸) → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
67		eqid 2761	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 = 𝐵
68	67	3mix2i 1353	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)
69	1	simp2d 1161	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ 𝑌)
70		eltpg 4647	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ 𝑌 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)))
71	69, 70	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)))
72	68, 71	mpbiri 261	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
73	72	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
74		eleq1 2849	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝐵 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
75	74	bicomd 226	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝐵 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
76	75	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
77	73, 76	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
78	42	bicomd 226	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ 𝑉))
79	41, 78	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ 𝑉))
80	79	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ 𝑉))
81	77, 80	mpbid 235	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → 𝑣 ∈ 𝑉)
82	81	ex 418	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑣 = 𝐵 → 𝑣 ∈ 𝑉))
83	82	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐵 → 𝑣 ∈ 𝑉))
84	83	impcom 413	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → 𝑣 ∈ 𝑉)
85		preq2 4695	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = 𝑣 → {𝐴, 𝐵} = {𝐴, 𝑣})
86	85	eleq1d 2846	. . . . . . . . . . . . . 14 ⊢ (𝐵 = 𝑣 → ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
87	86	eqcoms 2769	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝐵 → ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
88	87	biimpcd 252	. . . . . . . . . . . 12 ⊢ ({𝐴, 𝐵} ∈ 𝐸 → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ 𝐸))
89	88	ad2antrl 741	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ 𝐸))
90	89	impcom 413	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → {𝐴, 𝑣} ∈ 𝐸)
91	84, 90	jca 521	. . . . . . . . 9 ⊢ ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸))
92	91	ex 418	. . . . . . . 8 ⊢ (𝑣 = 𝐵 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
93		tpid3g 4733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∈ 𝑍 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
94	93	3ad2ant3 1153	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
95	1, 94	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
96	95	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 = 𝐶) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
97		eleq1 2849	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝐶 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
98	97	bicomd 226	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝐶 → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
99	98	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 = 𝐶) → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
100	96, 99	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 = 𝐶) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
101	79	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 = 𝐶) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ 𝑉))
102	100, 101	mpbid 235	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 = 𝐶) → 𝑣 ∈ 𝑉)
103	102	ex 418	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑣 = 𝐶 → 𝑣 ∈ 𝑉))
104	103	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐶 → 𝑣 ∈ 𝑉))
105	104	impcom 413	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → 𝑣 ∈ 𝑉)
106		preq2 4695	. . . . . . . . . . . . . . 15 ⊢ (𝐶 = 𝑣 → {𝐴, 𝐶} = {𝐴, 𝑣})
107	106	eleq1d 2846	. . . . . . . . . . . . . 14 ⊢ (𝐶 = 𝑣 → ({𝐴, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
108	107	eqcoms 2769	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝐶 → ({𝐴, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
109	108	biimpcd 252	. . . . . . . . . . . 12 ⊢ ({𝐴, 𝐶} ∈ 𝐸 → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ 𝐸))
110	109	ad2antll 742	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ 𝐸))
111	110	impcom 413	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → {𝐴, 𝑣} ∈ 𝐸)
112	105, 111	jca 521	. . . . . . . . 9 ⊢ ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸))
113	112	ex 418	. . . . . . . 8 ⊢ (𝑣 = 𝐶 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
114	92, 113	jaoi 871	. . . . . . 7 ⊢ ((𝑣 = 𝐵 ∨ 𝑣 = 𝐶) → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
115	114	com12 33	. . . . . 6 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣 = 𝐵 ∨ 𝑣 = 𝐶) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
116	66, 115	impbid 215	. . . . 5 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸) ↔ (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
117	116	abbidv 2827	. . . 4 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → {𝑣 ∣ (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)} = {𝑣 ∣ (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)})
118		df-rab 3414	. . . 4 ⊢ {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸} = {𝑣 ∣ (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)}
119		dfpr2 4605	. . . 4 ⊢ {𝐵, 𝐶} = {𝑣 ∣ (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)}
120	117, 118, 119	3eqtr4g 2821	. . 3 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸} = {𝐵, 𝐶})
121	40, 120	eqtrd 2796	. 2 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶})
122	36, 121	impbida 813	1 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2739   ≠ wne 2956  {crab 3413  {cpr 4586  {ctp 4588  ‘cfv 6538  (class class class)co 7420  Vtxcvtx 29574  Edgcedg 29625  USGraphcusgr 29730   NeighbVtx cnbgr 29913

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-dju 9982  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-n0 12607  df-xnn0 12680  df-z 12694  df-uz 12966  df-fz 13640  df-hash 14475  df-edg 29626  df-upgr 29660  df-umgr 29661  df-usgr 29732  df-nbgr 29914

This theorem is used by:  nb3grpr  29963  nb3grpr2  29964  nb3gr2nb  29965